This paper provides a framework to evaluate the performance of single and double integrator networks over arbitrary directed graphs. Adopting vehicular network terminology, we consider quadratic performance metrics defined by the L2-norm of position and velocity based response functions given impulsive inputs to each vehicle. We exploit the spectral properties of weighted graph Laplacians and output performance matrices to derive a novel method of computing the closed-form solutions for this general class of performance metrics, which include H2-norm based quantities as special cases. We then explore the effect of the interplay between network properties (such as edge directionality and connectivity) and the control strategy on the overall network performance. More precisely, for systems whose interconnection is described by graphs with normal Laplacian L, we characterize the role of directionality by comparing their performance with that of their undirected counterparts, represented by the Hermitian part of L. We show that, for single-integrator networks, directed and undirected graphs perform identically. However, for double-integrator networks, graph directionality -expressed by the eigenvalues of L with nonzero imaginary part- can significantly degrade performance. Interestingly, in many cases, well-designed feedback can also exploit directionality to mitigate degradation or even improve the performance to exceed that of the undirected case. Finally we focus on a system coherence metric -aggregate deviation from the state average- to investigate the relationship between performance and degree of connectivity, leading to somewhat surprising findings. For example, increasing the number of neighbors on a ω-nearest neighbor directed graph does not necessarily improve performance. Similarly, we demonstrate equivalence in performance between all-to-one and all-to-all communication graphs.
more »
« less
Stable motion and distributed topology control for multi-agent systems with directed interactions
In this paper, we study stable coordination in multi- agent systems with directed interactions, and apply the results for distributed topology control. Our main contribution is to extend the well-known potential-based control framework orig- inally introduced for undirected networks to the case of net- works modeled by a directed graph. Regardless of the particular objective to be achieved, potential-based control for undirected graphs is intrinsically stable. Briefly, this can be explained by the positive semidefiniteness of the graph Laplacian induced by the symmetric nature of the interactions. Unfortunately, this energy finiteness guarantee no longer holds when a multi-agent system lacks symmetry in pairwise interactions. In this context, our contribution is twofold: i) we formalize stable coordination of multi-agent systems on directed graphs, demonstrating the graph structures that induce stability for a broad class of coordination objectives; and ii) we design a topology control mechanism based on a distributed eigenvalue estimation algorithm to enforce Lyapunov energy finiteness over the derived class of stable graphs. Simulation results demonstrate a multi-agent system on a directed graph performing topology control and collision avoidance, corroborating the theoretical findings.
more »
« less
- Award ID(s):
- 1657235
- NSF-PAR ID:
- 10055505
- Date Published:
- Journal Name:
- Decision and Control (CDC), 2017 IEEE 56th Annual Conference on
- Page Range / eLocation ID:
- 3455 to 3460
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
This paper describes a group-level classification of 14 patients with prefrontal cortex (pFC) lesions from 20 healthy controls using multi-layer graph convolutional networks (GCN) with features inferred from the scalp EEG recorded from the encoding phase of working memory (WM) trials. We first construct undirected and directed graphs to represent the WM encoding for each trial for each subject using distance correlation- based functional connectivity measures and differential directed information-based effective connectivity measures, respectively. Centrality measures of betweenness centrality, eigenvector centrality, and closeness centrality are inferred for each of the 64 channels from the brain connectivity. Along with the three centrality measures, each graph uses the relative band powers in the five frequency bands - delta, theta, alpha, beta, and gamma- as node features. The summarized graph representation is learned using two layers of GCN followed by mean pooling, and fully connected layers are used for classification. The final class label for a subject is decided using majority voting based on the results from all the subject's trials. The GCN-based model can correctly classify 28 of the 34 subjects (82.35% accuracy) with undirected edges represented by functional connectivity measure of distance correlation and classify all 34 subjects (100% accuracy) with directed edges characterized by effective connectivity measure of differential directed information.more » « less
-
Markov Chain Monte Carlo (MCMC) has been the de facto technique for sampling and inference of large graphs such as online social networks. At the heart of MCMC lies the ability to construct an ergodic Markov chain that attains any given stationary distribution \pi, often in the form of random walks or crawling agents on the graph. Most of the works around MCMC, however, presume that the graph is undirected or has reciprocal edges, and become inapplicable when the graph is directed and non-reciprocal. Here we develop a similar framework for directed graphs called Non- Markovian Monte Carlo (NMMC) by establishing a mapping to convert \pi into the quasi-stationary distribution of a carefully constructed transient Markov chain on an extended state space. As applications, we demonstrate how to achieve any given distribution \pi on a directed graph and estimate the eigenvector centrality using a set of non-Markovian, history-dependent random walks on the same graph in a distributed manner.We also provide numerical results on various real-world directed graphs to confirm our theoretical findings, and present several practical enhancements to make our NMMC method ready for practical use inmost directed graphs. To the best of our knowledge, the proposed NMMC framework for directed graphs is the first of its kind, unlocking all the limitations set by the standard MCMC methods for undirected graphs.more » « less
-
We propose and analyze a new stochastic gradient method, which we call Stochastic Unbiased Curvature-aided Gradient (SUCAG), for finite sum optimization problems. SUCAG constitutes an unbiased total gradient tracking technique that uses Hessian information to accelerate convergence. We analyze our method under the general asynchronous model of computation, in which each function is selected infinitely often with possibly unbounded (but sublinear) delay. For strongly convex problems, we establish linear convergence for the SUCAG method. When the initialization point is sufficiently close to the optimal solution, the established convergence rate is only dependent on the condition number of the problem, making it strictly faster than the known rate for the SAGA method. Furthermore, we describe a Markov-driven approach of implementing the SUCAG method in a distributed asynchronous multi-agent setting, via gossiping along a random walk on an undirected communication graph. We show that our analysis applies as long as the graph is connected and, notably, establishes an asymptotic linear convergence rate that is robust to the graph topology. Numerical results demonstrate the merits of our algorithm over existing methods.more » « less
-
Graphs are the dominant formalism for modeling multi-agent systems. The algebraic connectivity of a graph is particularly important because it provides the convergence rates of consensus algorithms that underlie many multi-agent control and optimization techniques. However, sharing the value of algebraic connectivity can inadvertently reveal sensitive information about the topology of a graph, such as connections in social networks. Therefore, in this work we present a method to release a graph’s algebraic connectivity under a graph-theoretic form of differential privacy, called edge differential privacy. Edge differential privacy obfuscates differences among graphs’ edge sets and thus conceals the absence or presence of sensitive connections therein. We provide privacy with bounded Laplace noise, which improves accuracy relative to conventional unbounded noise. The private algebraic connectivity values are analytically shown to provide accurate estimates of consensus convergence rates, as well as accurate bounds on the diameter of a graph and the mean distance between its nodes. Simulation results confirm the utility of private algebraic connectivity in these contexts.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1109/CDC.2017.8264165
